JOURNAL OF GUIDANCE, CONTROL, AND DYNAMICS

Vol. 26, No. 3, May–June 2003

Optimal Trajectory Planning for Hot-AirBalloons in Linear Wind Fields

Tuhin Das¤ and Ranjan Mukherjee†

Michigan State University, East Lansing, Michigan 48824-1226and

Jonathan Cameron‡

Jet Propulsion Laboratory, California Institute of Technology, Pasadena, California 91109

The altitude of hot-air balloons is controlled by heating the air trapped inside the balloon and allowing the airto cool naturally. Apart from controlling the altitude, it is desirable to utilize the wind � eld to position the balloonat a target location while minimizing fuel consumption. This can be posed as an optimal control problem withfree end states, where the heat input to the system is the control variable. The problem is intractable becauseof the switching nature of the heat input and highly nonlinear state equations derived from the thermodynamicmodel of the balloon. In this paper we address the optimal control problem within a space of a few kilometerswhere we assume the wind � elds to be known and linear. We simplify the dynamic model of the balloon andobtain optimal trajectories to the target location by solving a two-point boundary-value problem. By re� ning thesimpli� ed dynamic model, the accuracy of the optimal trajectories are improved to match well with trajectoriesobtained using the nonlinear model. Our approach based on simpli� cation of the balloon dynamic model enablesus to solve the intractable nonlinear optimal control problem and provides insight into the optimal trajectories,such as number of switchings of input and loss of accuracy for speci� c wind pro� les. Except for these speci� c windpro� les, our approach yields accurate trajectories for the balloonand provides a solution to an important problemthat has not been adequately addressed in the literature.

NomenclatureA = cross-sectionalarea of sphere with volume

equivalent to the balloon, m2

CD = coef� cient of dragC Hfa = convective heat-transfercoef� cient between balloon

� lm and ambient air, W/m2KC Hgf = convective heat-transfercoef� cient between balloon

� lm and balloon gas (air), W/m2KCm = virtual mass coef� cientc f = speci� c heat of the balloon � lm, J/kg Kcpg = speci� c heat of the balloon gas (air), 1004:5 J/kg KG = solar constant, 1395:0 W/m2

Gra = Grashof number for convection between balloon� lm and ambient air

Grg = Grashof number for convection between balloon gas(air) and balloon � lm

g = acceleration caused by gravity, 9:81 m/s2

Ka = thermal conductivityof ambient airKg = thermal conductivityof balloon gas (air)Ma = molecular weight of air, 28.96 kg/kmolm f = mass of balloon � lm, kgm� = mass of fuel, kgmg = mass of balloon gas, kgm tot = total mass of balloon system, kgNua = Nusselt number for convection between

balloon � lm and ambient airNug = Nusselt number for convection between

balloon gas (air) and balloon � lm

Received 12 July 2003; revision received 22 January 2003; accepted forpublication30 January 2003.Copyright c° 2003by the American InstituteofAeronautics and Astronautics, Inc. All rights reserved. Copies of this papermay be made for personal or internal use, on condition that the copier paythe $10.00 per-copy fee to the Copyright Clearance Center, Inc., 222 Rose-wood Drive, Danvers, MA 01923; include the code 0731-5090/03 $10.00 incorrespondence with the CCC.

¤Graduate Student, Department of Mechanical Engineering.†Associate Professor, Department of Mechanical Engineering, 2555 En-

gineering Building.‡Senior Engineer, 4800 Oak Grove Drive. Senior Member AIAA.

Pr = Prandtl number of air, 0.72pa = ambient air pressure, PaPq f = net heat � ux to balloon � lm, WPqg = net heat � ux to balloon gas, WR = universal gas constant, 8314:3 J/kmol KNR = radius of a sphere with volume equivalent

to the balloon, mRe = Reynolds numberre = Earth relectivity (albedo), 0.18 (assuming

no cloud cover)rw = re� ectivity of the balloon � lm in the infrared spectrumrw sol = re� ectivity of the balloon � lm to solar radiationS = balloon surface area, equivalent to 4:835976V2=3, m2

Ta = ambient air temperature,KTBB = blackball temperature, 214.4 K (assuming

no cloud cover)T f = balloon � lm temperature, KTg = balloon gas temperature, Kt = time, sU = wind velocity in x direction, m/su = rate of heat input, WV = balloon volume, m3

V = wind velocity in y direction, m/sx = x coordinate of balloon, my = y coordinate of balloon, mz = balloon altitude, m®g = absorptivity of balloon gas (air) to solar

radiation, 0.003®geff = effective solar absorptivity of radiation®w = absorptivity of balloon � lm in the infrared spectrum®weff = effective solar absorptivity of the balloon � lm²g = emmisivity of balloon gas in the infrared spectrum²geff = effective infrared emmisivity of balloon gas²int = effective interchange infrared emmisivity²w = emmisivity of balloon � lm in the infrared spectrum²weff = effective infrared emmisivity of balloon � lm¹a = viscosity of the ambient air, Pa s¹g = viscosity of the balloon gas (air), Pa s½a = density of air, kg/m3

416

DAS, MUKHERJEE, AND CAMERON 417

½g = density of balloon gas (air), kg/m3

¾ = Stefan–Boltzmann constant, 5:669 £ 10¡8 W/m2K4

¿w = transmissivity of balloon � lm in the infrared spectrum¿w sol = transmissivity of the balloon � lm to solar spectrum

I. Introduction

H OT-AIR balloons are simple and relatively inexpensive aerialvehicles that have historically been used for scienti� c experi-

ments, such as precursor to manned space � ight, and astronomicaland telecommunications research. Its current applications includeaerial surveying, probing of the upper atmosphere to provide valu-able data to aircrafts, military applications,and recreation. In manyof these applications, trajectory control of the balloons is impor-tant and requires proper mathematical modeling of the mechanicaland thermal dynamics of the balloon. Although human operatorssuccessfully control balloons without using mathematical models,modeling will enable autonomous and semiautonomous operationwith higher fuel ef� ciency and motion accuracy and enable trajec-tories that might not be conceived by human operators.

The mathematical modeling of balloons beyond simple buoy-ancy calculations has been largely driven by high-altitude balloon� ights. Some of the early work on modeling was done by Kreider,1

Kreith and Kreider,2 and Carlsonand Horn.3 Their models took intoaccount thermodynamic in� uences of solar and infrared radiation,as well as optical/infrared absorptivity and related radiative prop-erties of balloon � lms. This is important for long � ight durationsand day–night transitions. The importance of vertical drafts nearthe surface for predicting ascent and descent motion of balloonswas established by Wu and Jones.4 The concept of buoyancy con-trol using a phase change � uid was analyzed and demonstrated byWu and Jones4 and Scheid et al.5 The primary lift in this system isprovided by a classic lighter-than-airballoon. The overall lift forceis modulated by a second balloon � lled with a phase change � uidthat remains gaseous near the ground and cools off and eventuallycondenses as the balloon goes up in the atmosphere.

Although thermodynamic models of balloons have been studiedearlier to address the altitudecontrolproblemand recentlytrajectorycontrol of balloons with lift-generating devices have been investi-gated (Aaron et al.6), � ight control of balloons in the presence oflateral wind � elds has not been reported. In this paper we addressthe trajectorycontrolproblemfor hot-airballoonssuch that they canreach a target location by controlling their altitude and riding thewind � eld judiciously. A balloon typically gains height when thetrapped air is heated by burning fuel. The buoyant force decreases,and the balloon starts descending when the trapped air cools natu-rally through heat exchange with the atmosphere.The control inputof a hot-airballoonis thereforeunidirectionaland switchesbetweenon and off states.We design the input to minimize a weightedsum ofthe total fuel consumed and the error in the � nal coordinatesof theballoon. This is motivated by the fact that very precise positioningof the balloon is not required for most applications.We assume thewind � eld to be known and linear and do not specify the time to betaken by the balloon to reach its destination. The knowledge of thewind � eld is justi� ed by the presenceof existingweatherdata and itslinearity warranted from extrapolation of data over short distances(few kilometers).Also, in the current environmentwhere formation� ying (Folta et al.7) is merited as a useful concept, knowledge ofwind data can be justi� ed by measurementsand sharingof data by aformation of balloons. Typically, wind data are statistical in nature,but in this paper we assume the data to be deterministic to keep theproblem simple.

This paper is organized as follows. In Section II we � rst intro-duce the thermal and dynamic models of hot-air balloons from theliterature. Based on certain assumptions and observations,we thenderive a linear model of the balloon for the purpose of control de-sign. The linear model enables us to cast the optimal control prob-lem as a tractable two-point boundary-valueproblem and providesvaluable insight into the optimal input-switching sequence, whichis discussed in Section III. In Section IV we � rst attempt to solvethe two-point boundary-value problem using a simple numerical

approach.Although this approach does not have good convergenceproperties, it provides useful information on the scale and magni-tude of the costate variables, which is a key to solving the optimaltrajectories numerically. We obtain optimal trajectories using themethod of relaxation (Press et al.8) and feed the input switchingsequence into the nonlinear model of the balloon for the purposeof comparison. The linear model is re� ned using results from thenonlinear model and the process repeated until the nonlinear modeland the relaxation algorithm yield matching trajectories. In mostsimulations we performed, a single iteration was suf� cient to re� nethe linearmodelandobtainaccurateoptimal trajectories.We presentsimulation results in Section V and provide concludingremarksandfuture research directions in Section VI.

II. Mathematical ModelA. Thermal and Dynamic Model

Based on thermalanddynamicmodels of balloonsbyCarlsonandHorn3 and the coordinatesystem descriptionin Fig. 1, the equationsof motion of hot-air balloonscan be described by the vertical force-balance equation

.m tot C Cm ½a V /d2z

dt2D g.½aV ¡ m tot/ ¡ 1

2½a ACD

dz

dt

­­­­dz

dt

­­­­(1)

the heat-balance equation for the balloon � lm

m f c fdT f

dtD Pq f (2)

and the heat-balance equation for the lifting gas (hot air)

mgcpgdTg

dtD Pqg ¡

³gmgTg

Ta

´dz

dtC u (3)

where

V Dmg RTg

pa Ma; m tot

1D mg C m f C m� ; A D ¼ NR2 (4)

The heat-� ux terms Pq f and Pqg in Eqs. (2) and (3) can be expressedas

Pq f D£G®weff.1=4 C re=2/ C ²int¾

¡T 4

g ¡ T 4f

¢C C Hgf.Tg ¡ T f /

C C Hfa.Ta ¡ T f / ¡ ²weff¾ T 4f C ²weff¾ T 4

BB

¤S (5)

Pqg D£G®geff.1 C re/ ¡ ²int¾

¡T 4

g ¡ T 4f

¢¡ C Hgf.Tg ¡ T f /

¡ ²geff¾ T 4g C ²geff¾ T 4

BB

¤S (6)

Fig. 1 Trajectory of a hot-air balloon in a wind � eld.

418 DAS, MUKHERJEE, AND CAMERON

where

®weff D ®w

µ1 C

¿wsol.1 ¡ ®g/

1 ¡ rwsol.1 ¡ ®g/

¶; ®geff D

®g¿wsol

1 ¡ rwsol.1 ¡ ®g/

²int D ²g²w

1 ¡ rw.1 ¡ ²g/; ²geff D ²g¿w

1 ¡ rw.1 ¡ ²g/

²weff D ²w

µ1 C

¿w.1 ¡ ²g/

1 ¡ rw.1 ¡ ²g/

¶(7)

and S D 4¼ NR2 . The expressions for ²g , CH fa, and CHgf are given inthe Appendix. The motion of the balloon in the x and y directionsis caused by wind drag and can be approximated by the equations

Px D ° U ; Py D ° V (8)

where ° is the ratio of the balloonspeedand the absolutewind speedand is a measure of drag. Because our wind � eld is linear, U and Vcan be described by the equations

U D U 0 C k11.x ¡ x0/ C k12.y ¡ y0/ C k13.z ¡ z0/ (9a)

V D V 0 C k21.x ¡ x0/ C k22.y ¡ y0/ C k23.z ¡ z0/ (9b)

where U 0 and V 0 are the x and y components of the wind velocityat the initial location of the balloon .x0; y0; z0/, and the parameterski j , i D 1; 2, j D 1; 2; 3 are constants. The velocity of the wind inthe vertical direction is assumed to be zero.

B. Model Simpli� cation for Control Problem FormulationThe intricately coupled thermal and dynamic equations of the

balloon in Eqs. (1–3), (5), (6), and (8) can be represented in thestandard state-space form

PX D f .X; u/ (10)

where X is the vector of state variables and f is a nonlinear vectorfunctionof the states and the input.Our goal is to obtain the optimalinput u that minimizes the cost funtional

J D Á[X.t f /] CZ t f

0

L.X; u/ dt (11)

where Á is a measure of the error in the Cartesian coordinates ofthe balloon at the � nal time t D t f . The term Á[X.t f /] is included inthe cost functionbecause a � xed end-stateproblemwill be ill posedwith the balloon having no direct control in the x and y directions.The Lagrangian function L was chosen as the fuel consumed by theballoon such that the cost function is a weighted sum of the totalfuel consumedand the terminal error. The equationsfor the costatesor adjoint variables can be written as (Lewis and Syrmos9)

P̧ T D ¡@ H

@X; ¸T .t f / D

³@Á

@X

´

t f

; H1D L C ¸T f (12)

where H is the Hamiltonian. The optimal input can be obtained byminimizing the Hamiltonian over admissible choices of the input,whichessentiallyhas on andoff states.This posesan extremelydif� -cult problemfor our nonlinearsystem in which the costateequationsare very complicated.To make the problemtractable,we make a fewassumptions, some of them based on observations, that essentiallyresult in linearizationof the dynamicand thermodynamicequations.These assumptions are discussed next:

1) The term .m tot C Cm½a V / in Eq. (1) is assumed to be constant.This assumptionis reasonablebecausethe fractionalchange in massof the balloon caused by consumptionof fuel and change in volumeis quite insigni� cant. The value of the constant is determined frominitial conditions.

2) The cross-sectional area of the balloon A is assumed to beconstant. This allows further simpli� cation of Eq. (1).

3) The variation in ambient temperature and pressure is assumedto be small over the rangeof travelof the balloon.Both Ta and pa aretherefore treated as constants.This implies that ½a is also constant.

4)We nowfocusourattentionon the drag termin Eq. (1) involvingPzjPzj. This term increases as the square of the velocity and thereforehas a limiting effect on the velocity of the balloon. Assuming theballoon velocity to satisy jPzj · Pzmax , we can approximate the dragterm using the method of least squares (Kreyszig10) by minimizingthe integral

I DZ CPzmax

¡Pzmax

.PzjPzj ¡ k Pz/2 dPz

with respect to k. This results in the approximation

PzjPzj ¼ k Pz; k1D 0:75Pzmax (13)

The equation used for modeling drag in x and y directions, namely,Eq. (8), is different from Eq. (1), which models drag in the z di-rection. We chose the simpler drag model in x and y directionssimply because it reduces the dimension of the problem by twostates and two costates. The choice of a simpler model is also justi-� ed by the fact that the balloon dynamics in the x and y directionsare much less complicated than the dynamics of the balloon in thez direction.

5) The most signi� cant and key assumption in this paper per-tains to linearizationof the heat-balanceequations,namely,Eqs. (2)and (3). It is evident from Eqs. (5) and (6) that the terms Pq f and. Pqg ¡ gmgTg Pz=Ta/ in Eqs. (2) and (3) are nonlinear functions of Pz,T f , and Tg . To express them in linear form, we express them as

¡Pq f D a1 Pz C b1T f C c1Tg (14a)

¡ Pqg C gmgTg Pz=Ta D a2 Pz C b2T f C c2Tg (14b)

and identify theconstantsa1 , b1, c1, a2 , b2 , andc2 fromsimulationre-sults using the method of least squares(Kreyszig10). The simulationresults indicate an excellent match between the left- and right-handsides of both Eqs. (14a) and (14b) and af� rm the accuracy of thelinear representation.

To demonstrate the advantages of linearization of the heat-� uxequations, we present results from two simulation maneuvers. Theparametersused in simulation are provided in Section V. In the � rstsimulation the balloon was commanded to hover at 12 km startingfrom an initial altitudeof 14 km. The hover was achieved by simplyswitching on the heat input when the balloon was below the hoveraltitude and had a downward vertical velocity. Figure 2a shows thatthe balloon initially drops below the hover altitude, then rises backup slowly,and � nallyhoversat 12 km. The least-squarescoef� cientswere obtained as

a1 D 0:236625 £ 104; a2 D 8:152573 £ 103

b1 D 0:282035 £ 104; b2 D ¡2:344357 £ 103

c1 D ¡0:257485 £ 104; c2 D 2:272179 £ 103

These coef� cients were substituted in Eq. (14) to obtain linear ap-proximation of the terms ¡Pq f and .¡Pqg C gmgTg Pz=Ta/; the resultsare shown in Figs. 2b and 2c in dashed lines. These plots are almostindistinguishable from the plots of ¡ Pq f and .¡Pqg C gmgTg Pz=Ta/obtained from the simulationof the nonlinearmodel, shown in solidlines.

We present results of one more simulation to be convincedof theaccuracy of linear approximation of the thermodynamic equations.In this simulation the heat input was switched on and off multipletimes in an arbitrarymanner.The balloonaltitudeis shown in Fig. 3a,and the plots of ¡ Pq f and .¡Pqg C gmgTg Pz=Ta/ are shown in Figs. 3band 3c.Theseplots onceagain indicate that the linearheat-� ux termsapproximate their actual variation very closely. The least-squares

DAS, MUKHERJEE, AND CAMERON 419

a)

b)

c)

Fig. 2 Least-squares � t for heat-� ux terms for a hover maneuver.

a)

b)

c)

Fig. 3 Least-squares � t for heat-� ux terms for another maneuver.

coef� cients were obtained as

a1 D 1:187972 £ 104; a2 D 8:179336 £ 103

b1 D 0:432366 £ 104; b2 D ¡2:501151 £ 103

c1 D ¡0:383053 £ 104; c2 D 2:414179 £ 103

III. Optimal Control FormulationA. Linear State-Space Representation

Based on the assumptionsmade in the preceding section,we cannow express the dynamical and thermodynamical equations of theballoon in the linear state-space form

PX D AX C Bu; X.0/ D X0 (15)

where the state vector X and matrix A are de� ned as

X1D

2

66666666666664

NxNyNzPNz

T f

V

x7

x8

x9

3

77777777777775

A1D

2

66666666666664

° k11 ° k12 ° k13 0 0 0 0 1 0

° k21 ° k22 ° k23 0 0 0 0 0 1

0 0 0 1 0 0 0 0 0

0 0 0 p1 0 p2 ¡1 0 0

0 0 0 ¡Na1 ¡ Nb1 ¡Nc1 0 0 0

0 0 0 ¡Na2 ¡ Nb2 ¡Nc2 0 0 0

0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0

3

77777777777775

;

B1D

2

66666666666664

0

0

0

0

0

R=pa Mgcpg

0

0

0

3

77777777777775

(16)

and X0 is the state vector at the initial time t D 0. In Eq. (16) the � rstthree states are de� ned as Nx 1D .x ¡ xd/, Ny 1D .y ¡ yd/, Nz 1D .z ¡ zd/,where xd , yd , zd are the desiredCartesiancoordinatesof the balloon,and the last three states are constants, given by the relations

x71D m totg=.m tot C Cm ½a V /; x8

1D ° . U 0 ¡ k11x0 ¡ k12 y0/

x91D ° . V 0 ¡ k21x0 ¡ k22 y0/ (17)

In Eq. (16) we also have p11D ¡0:5k½aCD A=.m tot C Cm ½a V /,

p21D g½a=.m tot C Cm½a V /, and

Na11D a1=m f c f ; Na2

1D a2 R=pamgcpg

Nb11D b1=m f c f ; Nb2

1D b2 R=pamgcpg

Nc11D c1 pa Ma=m f c f mg R; Nc2

1D c2=mgcpg (18)

where the constantsa1 , a2, b1 , b2 , c1, c2 were de� ned in Eq. (14). Inour model, the volume of the balloon V is one of the state variables.Because V and Tg are related by the algebraicexpressionin Eq. (4),Tg can be alternativelyused as the state variable instead of V .

To proceed with our analysis, we express the cost function inEq. (11) as follows:

J D 1

2XT .t f /FX.t f / C ¯

2

Z t f

0

u2 dt (19)

where F1D diag.1 1 1 0 0 0 0 0 0/ is a diagonal pos-

itive semide� nite matrix chosen to penalize the terminal error in theCartesian coordinates of the balloon, all of them equally, and ¯ is

420 DAS, MUKHERJEE, AND CAMERON

a positive scalar representing the fuel cost. The Hamiltonian andcostate equations can now be obtained from Eqs. (12) and (15) asfollows:

H D .¯=2/u2 C ¸T .AX C Bu/

D .¯=2/u2 C .¸6 R=pa Mgcpg/u C ¸T AX (20)

P̧ D ¡AT ¸; ¸.t f / D FX.t f / (21)

The optimal input is obtained by minimizing the Hamiltonian inEq. (20) for admissible choices of the input. If we assume u toswitch between values 0 and ´, where ´ is the heat added to theballoongas when the input is on, the optimal input can be expressedas follows:

u D»

0; if ¸6 R=.cpg pa Mg¯/ > ¡0:5´

´; if ¸6 R=.cpg pa Mg¯/ · ¡0:5´(22)

This indicates that the optimal input depends only on the trajectoryof the costate ¸6 , and switching occurs when

¸6 D ¡0:5´cpg pa Mg¯=R (23)

B. Insight into Optimal TrajectoriesIn the absence of linearization the states and costates are de-

scribed by coupled nonlinear differential equations, and it is notpossible to obtain an analyticalexpressionfor the optimal input. Bylinearizationof the balloon dynamics, we are able to get an analyti-cal expressionfor the optimal input, namely, Eq. (22), and representthe states and costates in a cascade form, where the costates dependon themselves and the states depend on both the state and costatevariables. This reduces the computational complexity of the two-point boundary-valueproblem and provides useful insight into theoptimal trajectories. To obtain this insight, we � rst observe that ¸6

depends only on the � rst six costates. Because the optimal inputdepends on ¸6, we only need to analyze the trajectories of the � rstsix costates. These trajectories can be described by the differentialequations and boundary conditions

2

66666664

P̧1

P̧2

P̧3

P̧4

P̧5

P̧6

3

77777775

D

2

6666664

¡° k11 ¡° k21 0 0 0 0¡° k12 ¡° k22 0 0 0 0¡° k13 ¡° k23 0 0 0 0

0 0 ¡1 ¡p1 Na1 Na2

0 0 0 0 Nb1Nb2

0 0 0 ¡p2 Nc1 Nc2

3

7777775

2

6666664

¸1

¸2

¸3

¸4

¸5

¸6

3

7777775;

2

6666664

¸1

¸2

¸3

¸4

¸5

¸6

3

7777775

t f

D

2

6666664

Nx.t f /

Ny.t f /

Nz.t f /

0

0

0

3

7777775(24)

The preceding costate equations can also be written asµ P̧

1

P̧2

¶D M1

µ¸1

¸2

¶; M1

1D ¡°

µk11 k21

k12 k22

¶

P̧3 D ¡° k13¸1 ¡ ° k23¸2

2

4P̧

4

P̧5

P̧6

3

5 D M2

2

4¸4

¸5

¸6

3

5 C

2

4¡1

0

0

3

5 ¸3; M21D

2

4¡p1 Na1 Na2

0 Nb1Nb2

¡p2 Nc1 Nc2

3

5

(25)

which indicates that the expression for ¸6 will have the form

¸6 D C1e®1 t C C2e®2t C C3e

®3t C C4e®4 t C C5e®5t (26)

if the eigenvalues of M1 and M2 are distinct. In Eq. (26) Ci and®i , i D 1, 2, 3, 4, 5 are arbitrary constants and eigenvalues of M1

and M2 , respectively. If all of the eigenvalues are additionally real,Eq. (26) implies that the switching condition in Eq. (23) can occur atmost four times (Leitmann11). The matrix M1 will have real eigen-values if there exist no vortices in the wind � eld. This might be areasonable assumptionbecause vortices can cause the balloon to betrapped in a � xed region. Because entries of M2 depend on physicalparameters of the balloon, M2 will have real eigenvalues depend-ing on values of these parameters. The knowledge of the maximumnumber of switchings provides insight into the balloon trajectoriesbecause one can now conjure up wind � elds that will cause the heatinput to switch one, two, three, or four times. If M1 and M2 haveimaginary eigenvalues, there could be more switchings—the num-ber of switchings will however be � nite because our problem has a� xed time.

IV. Solution of Two-Point Boundary-Value ProblemA. Simple Numerical Approach

The optimal control problem of the hot-air balloon, described byEqs. (15), (22), and (24), results in a two-pointboundary-valueprob-lem. Because it imposes constraints on the control input and doesnot admit a closed-form analytical solution like the linear quadraticregulator problem, it has to be solved numerically. We provide asimple numerical approachfor solving the boundary-valueproblemin this section.The approachis similar to the shootingmethod (Presset al.8) and is described by the following steps:

1) We makean initialguessof the � rst three statesat the � nal time,namely, Nx.t f /, Ny.t f /, Nz.t f /. This gives us the boundary conditionsfor the costates at t D t f , as shown in Eq. (24).

2) Integrate Eq. (24) backward in time to obtain the costate tra-jectories.

3) Determine the input switching sequence from the trajectoryof¸6 and Eq. (22).

4) Use this input pro� le to integrate Eq. (15) forward in time anddetermine the values of the � rst three states at the � nal time.

5) Compute the error between values of Nx.t f /, Ny.t f /, Nz.t f / ob-tained in step 4 and their values assumed in step 1. Make appropriatechanges in the assumed values using the method of steepest descent(Press et al.8) and repeat steps 1–4 until the error converges to zero.

The method just discussed is easy to implement but often reachesa local minima and therefore fails to converge. It nevertheless pro-vides us with a clear idea of the scale and magnitude of the costatevariables. This knowledge is critical in computing the optimal tra-jectoriesnumericallyusingthe relaxationmethod(Press et al.8). Therelaxation method, which is discussed in the next section, has beensuccessfullyused in solving other complex nonlinearaerospaceop-timal control problems, for example, see Ref. 12.

B. Solution by Relaxation MethodIn the relaxation method ordinary differential equations (ODEs)

are replacedby � nite differenceequations(FDEs) on a grid or meshof points that spans the domain of interest. When the problem in-volves N coupled � rst-order ODEs representedby FDEs on a meshof M points, there are N variables at each of the M mesh points.With N £ M variables altogether the method involves inverting anM N £ M N matrix, but the matrix takes a special block diagonalform that allows an economical inversion both in terms of timeand storage. The solution of the FDE problem starts with an initialguess for xn;k , n D 1; 2; : : : ; N , k D 1; 2; : : : ; M . Then the incre-ments 1xn;k are determined such that xn;k C 1xn;k is an improvedapproximation.This is doneby a � rst-orderTaylor-seriesexpansion,the details of which can be found in the book by Press et al.8 Aftereach iteration an average correction error is computed by summingthe absolute values of all corrections, weighted by a scale factorappropriate to each variable:

err D 1M £ N

NX

k D 1

NX

j D 1

1X . j; k/

scalv. j/

DAS, MUKHERJEE, AND CAMERON 421

where scalv is an array that contains the typical size of the stateand costate variables.The numericalmethod discussed in Sec. IV.Awas very useful in estimating the entries of scalv for the costatevariables,andour successwith the relaxationmethodcanbepartiallyattributed to it. The relaxation method convergeswhen the value oferr becomes less than a small preselected value, and we were ableto get convergence for all simulations that we attempted.

We now provide a summary of the steps taken in implementationof the relaxation algorithm:

1) The � rst step is to make a good initial guess of the parametersNa1, Nb1, Nc1 , Na2, Nb2, Nc2 in Eq. (18). Because these parameters are de-pendent on the heat input sequence, their correct values are initiallyunknown. A good � rst estimate of the parameter values is obtainedfrom a hover maneuver of the balloon in the neighborhood of itsinitial altitude.

2) The parameter values obtained in step 1 are used in the relax-ation algorithm to obtain the heat input switching sequence and thecorresponding trajectory of the balloon.

3) The input switchingsequencein step 2 is fed into the nonlinearmodel of the balloon and the resulting trajectory comparedwith thetrajectory obtained in step 2. If the two trajectoriesare very similar,the input switching sequence in step 2 is optimal.

4) If the balloon trajectories obtained in steps 2 and 3 are notsimilar, the linear model of the balloon needs to be re� ned. To thisend, we use results obtained from the nonlinear model in step 3 tobetter estimate the parameters in step 1. Steps 2, 3, and 4 are thenrepeateduntil the trajectoriesobtainedfrom the linear and nonlinearmodels match closely.

V. Simulation ResultsTo demonstratethe ef� cacy of the iterativealgorithmdiscussedin

Sec. IV.B, we present simulation results. The dynamic parametersof the balloon are assumed to be

mg D 600; m f D 200; m� D 40

CD D 0:5; Cm D 0:5 (27)

where the units are the same as thoseused in the Nomenclature.Thematerial of the balloon � lm is chosen to be polyethylene,which hasthe following properties:

c f D 2302:7; rw D 0:127; rwsol D 0:114; ®w D 0:001

²w D 0:031; ¿w D 0:842; ¿wsol D 0:885 (28)

In conformity with Eq. (9), the linear wind � eld is assumed to be

U 0 D 1:5; k11 D ¡0:001; k12 D 0:0; k13 D 0:005

V 0 D 0:2; k21 D 0:004; k22 D ¡0:001; k23 D 0:003

(29)

The velocityof the balloon in the x and y directionsis approximatedusingEq. (8)with ° D 0:8. The drag termin Eq. (13) is approximatedusing k D 3. This value of k conforms well with our simulationresultswhere theverticalspeedof theballoonsatis� es jPzj · Pzmax D 4.

The rate of heat input to the balloon gas is chosen to be´ D 110; 000,assuminga fuelmass � ow rateof 1 gm/s anda calori� cvalue of 110 kJ/gm for the fuel. The initial states and the desiredCartesian coordinates of the balloon are assumed to be

Nx D ¡2000; Ny D ¡3000; Nz D 0; PNz D 0; T f D 250:0

Tg D 300:0; xd D 0; yd D 0; zd D 13,000 (30)

As the � rst step of the algorithmdiscussed in Sec. IV.B, we performa hover maneuver of the balloon in the neighborhoodof its initial al-titude to obtaina good � rst guessof the parameters in Eq. (18). From

a)

b)

c)

Fig. 4 Hover maneuversimulationfor a good� rst estimate of the heat-� ux parameters.

this maneuver, shown in Fig. 4, the following heat-� ux parametersare obtained:

Na1 D 0:013469; Nb1 D 0:006851; Nc1 D ¡0:000599

Na2 D 0:139032; Nb2 D ¡0:042376 Nc2 D 0:003931

(31)

As the second step of our algorithm, we solve the two-pointboundary-valueproblem using the parameters in Eq. (31). The totaltime for simulation is t f D 1000 s. The time step or mesh size ischosen as h D 1 s, which results in 1000 mesh points. Because thenonzeroentries of the F matrix were chosen as unity, ¯ is a measureof the fuel cost relative to the cost of terminal error. The value of¯ was selected to be 10¡7 , which implies that the cost for continu-ous fuel consumption for the entire duration of � ight is consideredto be equivalent to a terminal error of Á D 106 , that is, an error of1000 m in one of the Cartesian coordinates of the balloon. The re-sults of simulation are shown in Fig. 5. The switching condition inEq. (23) is satis� ed for ¸6 D ¡318:385. It can be seen from Fig. 5athat switching occurs twice, at t1 D 97 and at t2 D 498. The inputpro� le, which can be computed from Eq. (22), is shown in Fig. 5b.

We undertake the third step of our algorithm by simulating themotion of the balloon based on its nonlinear model and the inputpro� le in Fig. 5b. The z trajectory of the balloon obtained fromthe nonlinear model is compared with the z trajectory obtained instep 2 of our algorithm.These trajectories,shown in Fig. 6, indicatethe need for re� nement of our linear dynamic model. To re� ne thelinear dynamic model, we improve our estimates of the heat-� uxparameters by using the z trajectory of the balloon obtained fromthe nonlinear model, shown in Fig. 6. This is based on step 4 of ouralgorithm, discussed in Sec. IV.B. The new parameter values aregiven here:

Na1 D 0:025350; Nb1 D 0:009433; Nc1 D ¡0:000803

Na2 D 0:146690; Nb2 D ¡0:040604; Nc2 D 0:003785

(32)

Clearly, these values are not signi� cantly different from the valuesin Eq. (31), which were used as the � rst estimates. It is also obviousfrom Fig. 7 that the new values provide a good linear approxima-tion of the heat-� ux expressions. In continuation of step 4 of our

422 DAS, MUKHERJEE, AND CAMERON

a)

b)

Fig. 5 Switching condition and heat input pro� le obtained from the� rst run of the relaxation method.

Fig. 6 Comparison of balloon z trajectories after the � rst run of therelaxation method.

a)

b)

Fig. 7 Least-squares � t for heat-� ux terms after the � rst run of therelaxation method.

a)

b)

Fig. 8 Switching condition and heat input pro� le obtained from thesecond run of the relaxation method.

algorithm, we run the relaxation algorithm with the new parametervalues in Eq. (32). This results in the new input pro� le for the bal-loon, which is shown in Fig. 8b. The input pro� le is obtained fromthe trajectory of ¸6 , shown in Fig. 8a. It is clear from Fig. 8 thatthe number of switchings have increased from two to four, and thenew switching times are t1 D 85, t2 D 579, t3 D 626, and t4 D 656.To study the effect of model re� nement, we simulate the nonlinearmodel of theballoonusing the inputpro� le in Fig. 8b.The resultsareshown in Fig. 9 in dotted lines. These results match very well withthe results obtained from the second run of the relaxation method,shown in solid lines in Fig. 9. Because there is a good match be-tween the trajectories, the input pro� le in Fig. 8b is consideredto beoptimal. The x , y, and z trajectories in Fig. 9 are therefore optimalwith � nal coordinates

x D ¡176:43; y D ¡357:43; z D 12977:46 (33)

The relaxation algorithm required 23 iterations and approxi-mately 38 s to converge on a 1996 Sun Ultra-1 machine. Subse-quently, re� nement of the balloon dynamic model based on a singleiteration of the heat-� ux parameters yielded matching trajectoriesfrom the linear and nonlinear models. We have performed manysuch simulations and have consistently found that few iterationsare required for convergence:most examples require one iteration,some require two, and we seldom require three or more iterations.In some cases our algorithm fails to provide convergence, but thisoccurs only for speci� c wind pro� les. Except for these wind pro-� les, our algorithmprovidesa systematic framework for solving theintractable optimal control problem.

We conclude this section with a discussion on the wind pro� lesthat render our algorithm inapplicable. Because the balloon is un-controllable in x and y directions, our algorithm attempts to � nda good match between the z trajectories of the balloon obtainedfrom the linear and nonlinear models. It is implicitly assumed thata good match in the z trajectories will result in a good match in thex trajectories and the y trajectories. However, in some cases this

DAS, MUKHERJEE, AND CAMERON 423

a)

b)

c)

Fig. 9 Comparisonofx, y, and z trajectories of the balloonafter secondrun of the relaxation method.

assumption is not valid, and small errors between the z trajectoriesresult in large differencesbetween the x and y trajectoriesobtainedfrom the linear and nonlinear models. To explain further, we useEqs. (8) and (9) to express the motion of the balloon in x and ydirections as follows:

Px D ° [ U 0 C k11.x ¡ x0/ C k12.y ¡ y0/ C k13.z ¡ z0/] (34a)

Py D ° [V 0 C k21.x ¡ x0/ C k22.y ¡ y0/ C k23.z ¡ z0/] (34b)

If we now denote the x , y, z trajectories of the balloon obtainedfrom the nonlinear model as reference trajectories xr , yr , zr , we canrewrite Eq. (34) with x , y, z replaced by xr , yr , zr . By subtractingthis equation from Eq. (34), we get

µ POxPOy

¶D °

µk11 k12

k21 k22

¶ µOxOy

¶C °

µk13

k23

¶Oz (35)

where Ox 1D .x ¡ xr /; Oy 1D .y ¡ yr /, and Oz 1D .z ¡ zr / are x , y, and ztrajectoriesof the balloonobtained from the linear model expressedrelative to the reference trajectories. It now becomes obvious thatthe x and y trajectories obtained from the linear model can divergefrom those obtained from the nonlinear model when eigenvaluesofthe 2 £ 2 matrix in Eq. (35) have positive real parts. On the contrary,when the matrix is Hurwitz, boundson Ox and Oy will be proportionalto theboundon Oz. This essentiallyimplies that a goodmatchbetweenthe z trajectoriesobtained from the linear and nonlinearmodels willensure a good match in the x trajectories and y trajectories.

VI. ConclusionA hot-air balloon is a complex dynamical system with unidirec-

tional control of its altitude. It does not have direct control of itsmotion in the horizontal plane and has to ride the wind � eld judi-ciouslyto move toward a target location.In this paperwe address theoptimal control problem with the objective of designing trajectoriesof the balloonthat minimize a weightedsum of its terminal error andfuel consumption. The problem is intractable because of the non-linear thermodynamicmodel of the balloonand switchingnature ofthe heat input. An analytical solution to the problem does not exist,and a numerical solutionis elusive.Using a nontraditionalapproachfor linearization, we simplify the balloon dynamic model and nu-merically solve the ensuing two-point boundary-valueproblem. Were� ne the simpli� ed model and using an iterative approach, which

requires very few iterations, obtain accurate optimal trajectories.Unfortunately, our algorithm fails to converge when eigenvaluesofthe wind � eld have positive real parts. This somewhat limits theusefulnessof our approach, but considering the intractable and elu-sive nature of the problem it achieves a modest level of success inour � rst attempt at problem resolution.A complete resolutionof theproblem will require additional work that allows us to plan trajec-tories with arbitrary wind � elds as well as wind � elds described bystatistical data. The long-term goal of our research is to study theoptimal control problem in other balloon systems, such as balloonsusing phase change � uids, that will � nd applications in planetaryexploration.

Appendix: Balloon Heat TransferThe convective heat-transfer coef� cients CHfa and CHgf in

Eqs. (5) and (6) can be expressed as follows:

C Hfa D Nua Ka=2 NR; C Hgf D Nug Kg=2 NR (A1)

where the thermal conductivities Ka and Kg are given by the rela-tions

Ka D 1:99 £ 10¡3

³T 1:5

a

Ta C 112:0

´

Kg D 1:99 £ 10¡3

³T 1:5

g

Tg C 112:0

´(A2)

In Eq. (A1) Nua and N ug are computed from the values of Gra ,Grg , Re, and Pr . Whereas Pr has a constant value of 0:72, Gra ,Grg , and Re vary according to the following relations:

Gra D g½2a

.T f ¡ Ta/ NR3

Ta¹2a

; Grg D g½2g

.Tg ¡ T f / NR3

Tg¹2g

Re D 2PzNR½a

¹a(A3)

where

½a Dpa Ma

RTa; ½g D

pa Ma

RTg

¹a D 1:49 £ 10¡6

³T 1:5

a

Ta C 112:0

´

¹g D 1:49 £ 10¡6

³T 1:5

g

Tg C 112:0

´(A4)

and

Ta D

8<

:

Tinv if z < zinv

Tsea ¡ 0:00651z if zinv · z < ztrop

Tstrat if z ¸ ztrop

pa D»

.8:966 ¡ 0:0002025z/5:256 if z < ztrop

ptrope.1:69¡0:000157 z/ if z ¸ ztrop(A5)

where standard values of Tinv , Tsea, Tstrat, zinv, ztrop, and ptrop are asfollows:

Tinv D 282:0; Tsea D 288:15; Tstrat D 214:4

zinv D 944:7; ztrop D 10769:0; ptrop D 23502:0

Using Eqs. (A3) and (A4), Nua and N ug can now be computed asfollows:

424 DAS, MUKHERJEE, AND CAMERON

Nua D max[.Nua/1; .Nua/2]

Nug D(

0:325.Grg Pr/13 if .Grg Pr / > 1:34681 £ 10¡8

2:5£2:0 C 0:6.Grg Pr/

14

¤if .Grg Pr/ · 1:34681 £ 10¡8

(A6)

where .N ua/1 and .N ua/2 are de� ned as

.Nua/1 D»

0:37Re0:6 if V < Vcr

0:74Re0:6 if V ¸ Vcr;

Vcr D 538000:0.N ua/2 D 2:0 C 0:6.Grg Pr /

14

(A7)

The emmissivity of the balloon gas in the infrared spectrum ²g ,appearing in Eq. (7), is given by the relation

²g D 0:169¡1:746 £ 10¡6Tg

¢0:8152(A8)

Equations (A1–A8), together with Eqs. (1–9) in the main body ofthe paper, are representativeof the nonlinear model of the balloon.

AcknowledgmentThe � rst two authors gratefully acknowledge the support pro-

vided by NASA Jet Propulsion Laboratory, JPL Contract 1216050,in conducting this research.

References1Kreider, J. F., “Mathematical Modeling of High Altitude Balloon Per-

formance,” AIAA Paper 75-1385, 1975.2Kreith, F., and Kreider, J. F., “Numerical Prediction of the Performance

of High Altitude Balloons,” National Center for Atmospheric Research,NCAR Technical Note, NCAR-IN/STR-65, Boulder, CO, 1971.

3Carlson, L. A., and Horn, W. J., “New Thermal and Trajectory Model forHigh-Altitude Balloons,” Journal of Aircraft, Vol. 20, No. 6, 1983, pp. 500–507.

4Wu, J. J., and Jones, J. A., “Performance Models for Reversible FluidBalloons,” AIAA Paper 95-1623, May 1995.

5Scheid, R. E., Heun, M. K., Cameron, J. M., and Jones, J. A., “Thermo-dynamics, Phase Change, and Mass Transfer in Oscillatory Balloon Systems(Aerobots),” AIAA Paper 96-1870, June 1996.

6Aaron, K. M., Heun, M. K., and Nock, K. T., “Balloon Trajectory Con-trol,” AIAA Paper 99-3865, Aug. 1999.

7Folta, D., Newman, L., and Gardner, T., “Foundations of FormationFlying for Mission to Planet Earth and the New Millennium,” AIAA Paper96-3645, July 1996.

8Press, W. H., Teukolsky, S. A., Vetterling, W. T., and Flannery,B. P., Numerical Recipes in C, Cambridge Univ. Press, New York, 1992,p. 762.

9Lewis, F. L., and Syrmos, V. L., Optimal Control, Wiley, New York,1995, p. 153.

10Kreyzig, E., Advanced Engineering Mathematics, Wiley, New York,1993, p. 1001.

11Leitmann, G., An Introduction to Optimal Control, McGraw–Hill, NewYork, 1966, pp. 46, 47.

12Lu, P., Sun, H., and Tsai, B., “Closed-Loop Endo-AtmosphericAscentGuidance,” AIAA Paper 2002-4558, Aug.